Explaining the Implied Volatility Skew from the Rational Speculation Perspective: Calibration on the Taiwan Stock Index Option Market Chen, Son-Nan Tsai, Hui-Huang National Chengchi University National Taipei University Abstract Two cornerstones in Behavioral Finance, the limits of arbitrage and investor psychology, can explain the formation of implied volatility skew existing in Taiwan stock index option (TXO) market well. Adopting the real-time data which exhibits the limits of arbitrage in the futures market and designing two speculation models which describe the trading behavior of the market maker in the option market, this study successfully calibrates out the market maker’s perceived volatility with a view to exhibit the similar pattern to the volatility asymmetry in spot market. The trading behavior of the market maker in the prevalence of positive feedback traders is based upon the argument of Destabilizing Rational Speculation of (De Long et al., 1990) that is well suitable to a market full of noise traders like the TXO market. one thing deserves to mention is that the calibration of our second speculation model, Shifted Speculation Model, on the implied volatility curve involving merely single volatility parameter shows that it is a self-consistent model, improving the often unjustly maligned defect of Black-Scholes Model and conforming to the market practice. Keywords: Volatility Skew; Rational Speculation; Market Maker; Stock Index Option; Option Pricing; 1 1 Introduction The tremendous success of (Black and Scholes, 1973) and (Merton,1973) is based on the no-arbitrage theory and the true engineering, so-called the dynamic hedging argument, to determines option prices which are irrelevant to investor demand. However, many empirical studies document the existence of well-known option-pricing puzzles – the expensiveness and the skew pattern of stock index options. These puzzles show that BSM can’t be self-consistent. The related literature is surveyed by (Bates, 2003) and concludes that demand-pressure can really affect option prices and that a new pricing approach by market makers is needed. This paper takes this challenge and tries to explain the above empirical patterns of index option prices from the industry organization in the market: hedgers, market makers and speculators. (Bollen and Whaley, 2004) asserts that the volatility skew is due to the demand of the put for hedging on the index under the inevitable reality of limits of arbitrage1. The demand for hedging also affects the index futures prices, as (Chen, Cuny and Haugen, 1995) describes. However, the demand of options seems not to increase monotonically as the moneyness decreases, and then this argument can’t explain the second puzzle properly. Further, whereas financial theory and empirical evidence suggest that derivatives markets exist to facilitate hedging, a more popular perception is that they serve as an instrument of speculation, and the demand for speculation also tend to be invited as the spot volatility increases. Although (Chatrath et al., 2002) describes that the dominance of commercial trading is typical for stock index futures markets, the industry organization in Taiwan market is totally different. The ratio of trade volume made by individual investors is over 70%. It is difficult to believe that the demand for hedging is large enough persistently to affect the option price because that the portfolio tilts of the individuals are more severe than that of the mutual fund managers. In addition there are no designated dealers or specialists in Taiwan markets. It therefore seems quite unreasonable to attribute the volatility skew of Taiwan stock index option (hereafter TXO) market to the demand of options for hedging without any hesitation. In the light of the above observation, this paper is motivated to study the volatility skew in TXO market from the viewpoint of speculation. 1 Capital constraint and agency problems (Shleifer and Vishny, 1997) or collateral requirement (Liu and Longstaff, 2004) can limit intermediaries such as market makers to hedge their positions perfectly. 2 The destabilizing rational speculation argument by (De Long et al., 1990) is suited to describe the operation of TXO market. Basically individual investors are commonly seen as the noise traders or irrational speculators; proprietary traders and foreign institutional investors are seen as de facto market makers. Arbitrage positions undertaken by rational speculators cannot be unlimited because of risk aversion, so noise traders can affect spot prices.2 Some kind of noise traders like positive feedback traders – buy when prices rise and sell when prices fall – behaviors on extrapolative expectation,3 e.g. extrapolating price changes. They successfully show that how the early trading made by rational speculators’ triggers positive-feedback trading, and then increase volatility about fundamentals. Their view of destabilizing rational speculation is motivated partly by George Soros’ own investment strategy – betting not on fundamentals, but on future crowd behavior. More interest is that this argument can account for the front-running by investment banks. To the best of our knowledge, this argument has yet applied to the stock index option market, not to mention the interactions between the stock market and its derivative markets. This is just the challenge which this paper tries to take on with the help of Behavioral Finance.4 As to the implementation of speculation, a complex pricing model is never a top priority. One claim, announced by (Derman and Wilmott, 2009), that BSM is already the consensus pricing model is because of its clearness and robustness. The clearness comes from a true engineering, i.e. dynamic hedge, and then shows the relationship between the implied volatility and future spot volatility under a friction-free market. The robustness means that it allows market players easily manipulate the only input, i.e. volatility, so as to arrive at the price deemed more appropriate by them. The second feature implies that BSM is useful for measuring the relative values among options and that the implied volatility contains the trader’s view on the future in real world, whilst the first one implies that implied 2 According to studies related on Taiwan option market like (Barber et al., 2006) and (Chang et al., 2009), only proprietary traders and foreign institutional investors earn money in the Taiwan option market. Individuals in total lose money. 3 Extrapolative expectation is verified with the experiment in (Andreassen and Kraus, 1988). Subjects with some training in economics and shown with authentic stock price patterns will reveal this phenomenon. 4 (Shleifer and Summers, 1990) has a similar but more detailed discussion on the so-called “the noise trader approach to Finance” from the arguments of both the “limits of arbitrage” and the “investor sentiment”. And these two arguments are exactly the two cornerstones of Behavioral Finance. 3 volatility may be not equal to the future spot volatility because of the transaction cost, but the innovations of these two kinds of volatilities should be the same or have similar patterns. (Korn and Wilmott, 1988), (“KW” for short) based on the same true engineering as BSM but releasing the requirement of dynamic hedge, provides a framework for hedging and speculating with options to show how the speculator evaluates an option. One thing that deserves notice is that BSM is a special case in this framework. This paper adopts the analysis of (De Long et al., 1990) and follows KW to design pricing models for measuring subject values of TXOs to rational speculators. Rational speculators in this paper are assumed to be market makers who utilize the directional information from the stock index futures and then determine their prospecting volatility of the stock index by referring some patterns in stock market. This is similar to bet on the future crowd behavior in De Long et al. The most possible pattern relating both direction and volatility of the stock market is the volatility asymmetry. We call the implied volatility calibrated from models about speculation the perceived volatility with a view for convenience. In opposite, the implied volatility from BSM, without directional information in it, should reflect the level of demand on option from noise traders. With the empirical calibration on the real-time data of TXO, the first finding in this paper is that while implied volatility from BSM violates the internal arbitrage,5 the perceived volatility with a view does not. Then this paper finds that the perceived volatility with a view can fit the skewed volatility curve well. This means that the perceived volatility with a view designed by us can measure the unique future spot volatility prospected by market makers and that the pricing models in this paper are self-consistent. Finally, from the skewed volatility curve, this paper calibrates out the reverse relation between the speculator’s expected future spot prices and his perceived volatilities of the underlying asset. This verifies our projection that the rational speculator may determine his perceived volatility with a view in option market by referring the volatility asymmetry in spot market. And this is the major contribution on this paper. For one thing it supplements the literature of investigating the whys and wherefores behind the volatility skew of the stock index option market from the supply side of option, for another it supports the literature of studying implied volatility and future portfolio returns, dating back to (Giot, 2005). 5 Free of internal arbitrage means that by put-call parity the volatility of put should be the same as that of call. The arbitrage in BSM is volatility arbitrage, i.e. deviation of the implied volatility and the future spot volatility. However, the future spot volatility is uncertain at the time of trade. 4 The remainder of this article is organized as follows. Section 2 takes the related literature review. Section 3 introduces the models extended by us for describing the behavior of the rational speculator.